Let’s illustrate the process with an example, the first word is cat and we add it to the trie. Since it doesn’t have a prefix, we continue. Second word is cats, we add it to the trie and check whether it has a prefix, and yes it does, the word cat. So we append the pair <’cats’, ‘s’> to the queue, which is the current word and the suffix. The third word is catsdogcats, we again insert it to the trie and see that it has 2 prefixes, cat and cats. So we add 2 pairs <’catsdogcats’, ‘sdogcats’> and <’catsdogcats’, ‘dogcats’> where the former suffix correspond to the prefix cat and the latter to cats. We continue like this by adding <’catxdogcatsrat’, ‘xdogcatsrat’> to the queue and so on. And here’s the trie formed by adding example the words in the problem definition:

After building the trie and the queue, then we start processing the queue by popping a pair from the beginning. As explained above, the pair contains the original word and the suffix we want to validate. We check whether the suffix is a valid or compound word. If it’s a valid word and the original word is the longest up to now, we store the result. Otherwise we discard the pair. The suffix may be a compound word itself, so we check if the it has any prefixes. If it does, then we apply the above procedure by adding the original word and the new suffix to the queue. If the suffix of the original popped pair is neither a valid word nor has a prefix, we simply discard that pair.

An example will clarify the procedure, let’s check the pair <’catsdogcats’, ‘dogcats’> popped from the queue. Dogcats is not a valid word, so we check whether it has a prefix. Dog is a prefix, and cats is the new suffix. So we add the pair <’catsdogcats’, ‘cats’> to the queue. Next time we pop this pair we’ll see that cats is a valid word and finish processing the word catsdogcats. As you can see, the suffix will get shorter and shorter at each iteration, and at some point the pair will either be discarded or stored as the longest word. And as the pairs are being discarded, the length of the queue will decrease and the algorithm will gradually come to an end. Here’s the complete algorithm: